Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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0
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1answer
47 views

compute $\nabla f$ for a function over a cone

Let $D$ be the cone $D=\{rt:r>0, t\in\Omega\}$ with $\Omega\subset S^{n-1}$. I want to show that $$ \frac1{r^2}\int_{B_r}\frac{|\nabla f(x)|^2}{|x|^{n-2}} dx= C(n,g)r^{2(a-1)} $$ where $C(n,g)$ is ...
0
votes
0answers
12 views

What conditions are needed for cpt support of anti-gradient?

I'm reading through a paper by Michael Christ, "On the $\bar{\partial}_b$ Equation for Three-Dimensional CR Manifolds" found in the Proceedings of Symposia in Pure Mathematics, Volume 52, Part 3. One ...
0
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1answer
19 views

How are the following inequalities concluded based on this first one?

$$I-\frac{\epsilon}{3} \leq s(f,T) \leq \underline{I} \leq \overline{I}\leq S(f,T) \leq I+ \frac{\epsilon}{3}$$ from this, the following is concluded, but how? $$1.\ \ \ 0 \leq |I-\underline{I}|\leq ...
2
votes
1answer
31 views

Combining Fubini and Tonelli's in one single Assumption

I am referring to the statements on Wikipedia, there it is said that Fubini's Theorem states that if $f : X\times Y \to \mathbb R$ is integrable, then $$ \int_X \left( \int_Y f(x,y) dy\right) dx = ...
0
votes
1answer
36 views

How to find the image of an arbitrary element under this operator?

Let $(e_n)$ be a total orthonormal sequence in a separable Hilbert space $H$ and define the right shift operator to be the linear operator $T \colon H \to H$ such that $T e_n = e_{n+1}$ for $n = 1, 2, ...
1
vote
1answer
22 views

Partial Integration for measures

I have the following formula in mind, $\mu$ a measure on $\mathbb{R}$. Any sigma-finite measure on $\mathbb{R}$ can be decomposed into a absolut continuous part, a "point measure" and a singular ...
4
votes
2answers
54 views

How can something be proved unsolvable? [duplicate]

My question specifically deals with certain real indefinite integrals such as $$\int e^{-x^2} {dx} \ \ \text{and} \ \ \int \sqrt{1+x^3} {dx}$$ Books and articles online have only ever said that these ...
0
votes
1answer
21 views

Some difficile question about almost everywhere valid properties

Let $\mu$ be a measure and $[f]\in L^2(\mu)$, i.e. $$[f]=\left\{g\in\mathcal{L}^2(\mu):f\equiv g\;\;\;\mu\text{-almost everywhere}\right\}$$ Moreover, let $x^+:=\max(x,0)$ for $x\in\mathbb{R}$. ...
0
votes
0answers
6 views

Find the coordinates of the center of mass of the waved line ABC if A(1,1),B(1,2),C(0,0) and density is p(x,y)=y

Find the coordinates of the center of mass of the waved line ABC if A(1,1),B(1,2),C(0,0) and density is p(x,y)=y. I don't know much about that kind of problems,I didn't find appropriate information on ...
2
votes
1answer
63 views

Is this one infinite dimensional manifold?

First of all, just to give context to the question: I've been reading some articles in Physics, and those articles imply without proof that one space is one infinite dimensional manifold. One of those ...
2
votes
1answer
30 views

Eigenvalue of Laplacian with Robin boundary condition

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with smooth boundary $\partial \Omega$ and let $\nu$ denote the outer unit normal. Let $u$ be an eigenfunction of $-\Delta$ in $\Omega$ satisfying ...
0
votes
1answer
33 views

Why is the following true? functions

$$x , x_0 \in [a,b]$$ $x_0$-fixed $f \in D(a,b)$- differentiable on [a,b] $$\triangle (x)=f(x)-f(x_0)-f'(x_0)(x-x_0)$$ $$\triangle '(x)=f'(x)-f'(x_0)$$
1
vote
1answer
30 views

Find min/max values of $f(x,y)=x^3-y^3+3xy$ in the set $K=[0,4]\times[-4,0]$

Find the biggest and the smallest values of the function $f(x,y)=x^3-y^3+3xy$ in the set $K=[0,4]\times[-4,0]$. So using partial derivatives we find that the critical points are $(0,0)$ and $(1,-1)$. ...
0
votes
1answer
50 views

rigorous proof in first year multivariable calculus

Hi could anyone help producing a proof for the following? If $f(x,y)$ is continuous on a closed and bounded region $R$, then $f$ has both an absolute maximum and an absolute minimum on $\mathbb{R}$. ...
4
votes
2answers
39 views

How do I find $\|T\|$ when given a matrix $T$?

How do I find the norm $\|T\|$ of T: $\mathbb{R}^3 \rightarrow \mathbb{R}^3$ is defined by $T(x) := Ax$, where $A:= \begin{pmatrix} 0 & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 3 ...
-1
votes
2answers
85 views

Fundamental Matrix (Floquet theory)

Let $\begin{pmatrix} \dot{x}_1 \\\dot{x}_2\end{pmatrix}=A(t)\begin{pmatrix}x_1\\x_2 \end{pmatrix}$ where $$A(t)=\begin{pmatrix}\alpha(t)+\cos(t)&\sin(t)\\ -\sin(t)& ...
2
votes
1answer
108 views

Is it feasible for a sophomore in high school (15 years old) to learn complex analysis? [closed]

I've been reading up on complex analysis and it seems an incredibly fascinating subject to me and one I'd like to learn more about. However, most of the books I've come across are for graduates, which ...
3
votes
0answers
54 views

Homework problem about the smallest sigma algebra:

Let $\mathscr{E} \subset 2^X $, then there is a unique smallest $\sigma$-algebra containing $\mathscr{E} $. Proof(Attempt): Since $\mathscr{E} \subset 2^X $, $2^X$ is $\sigma$-algebra containing ...
0
votes
3answers
87 views

Locally Lipschitz and Gâteaux Derivative if and only if Frechet Derivative

Consider $f$ locally Lipschitz. So $f$ is Gâteaux Derivative if and only if $f$ is Frechet Derivative. PS.: the converse is trivial.
2
votes
3answers
90 views

Looking for an example of an increasing function $f:[a,b] \to [a,b]$ which is discontinuous at infinitely many points

I am looking for an example of an increasing function $f:[a,b] \to [a,b]$ which is discontinuous at infinitely many points ; please help , thanks in advance .
5
votes
1answer
40 views

Prove that $\exists k\in \mathbb{N}^*$ such that $\|a-a_k\|<\varepsilon$

Let $E$ be a normed linear space, $C$ compact and $f:C\to C$ a function such that $\|f(x)-f(y)\|\geq \|x-y\|$ for all $x,y\in C$. Then $f$ is an isometry. Note: I'm having trouble trying to prove ...
0
votes
1answer
43 views

Bound of Mann iterative sequence

There is theorem in the book of Charles Chidume "Geometric Properties of Banach Spaces and Nonlinear Iterations" My question is: why if the underlined conditions are satisfied {Xn} is bounded (proof ...
1
vote
1answer
50 views

Is my approach right? Or is there a better way?

Let $E,F$ normed linear spaces, let $C$ connected of $E$, $D\subset F$, and $f:C\to D$ such that $f$ is open (i.e. sends open sets in $C$ "which is the same as open sets of $E$ intersected with ...
2
votes
1answer
47 views

1 parameter subgroups and Lie groups

I was just reading some lectures notes (that are not online available unfortunatley) on Lie groups and found that sometimes the author just says if he wants to prove something for all Lie group ...
3
votes
0answers
19 views

Analysis, area of rotating graph around x axis.

The proof comes to the following part which doesn't make a lot of sense for me: $$S(P)= 2 \pi \sum_{i=1}^{n} f(\epsilon_i) \sqrt{1+f'^2(\epsilon_i)}\triangle x_i + 2 \pi \sum_{i=1}^{n} ...
1
vote
2answers
78 views

Showing that $f$ is $C^\infty$

Question: Let $f: U \to \mathbb R$ be a continuous function, with $U \subset \mathbb R^2$ open, such that $$(x^2 +y^4)f(x,y) + f(x,y)^3 = 1,\, \,\, \forall (x,y) \in U$$ Show that $f$ is of ...
-4
votes
1answer
49 views

When the logarithm and integral can be commuted? [closed]

When the Logarithm and integral can be commuted?
0
votes
1answer
29 views

Riemann-integral of a non-continuous function

Let $f : \,\mathbb R \to \mathbb R$ be a function with a discontinuity at point $x_0$. How can I prove formally that $f$ the Riemann-integral of $f$ exists, i.e. that $f$ fulfills $\sum_k (\sup ...
2
votes
0answers
13 views

Integral over $B_1^n(0)$

Evaluate $I = \int_{B_1^n(0)} (a_1x_1 + \cdots + a_nx_n)^{2/3}$ where $a_j \in \mathbb{R}$. Here's where I'm at, following a hint: Consider an orthonormal transformation $T$ with the first row equal ...
1
vote
0answers
22 views

Boundary of surface

Let $S$ be the region in $\mathbb{R}^2$ bounded by $x$-axis, $x=1$, and $y=x$. Define $$ f(x,y) = \begin{cases} 0 &\mbox{if } x = 0 \text{ or if $x$ or $y$ is irrational} \\ 1/q & \mbox{if ...
4
votes
2answers
73 views

$a_n\downarrow 0, \sum\limits_{n=1}^{\infty}a_n=+\infty, b_n=min\{a_n,1/n\}$, prove $\sum b_n $ diverges.

$a_n\downarrow 0, \sum\limits_{n=1}^{\infty}a_n=+\infty, b_n=min\{a_n,1/n\}$, prove $\sum b_n $ diverges. In fact, I have known that two positive divergent series $\sum a_n ~\sum b_n$, ...
0
votes
1answer
33 views

Hilbert space isometric to a subspace of its dual

Let $\cal H$ be a Hilbert space, and let $\cal H^\ast$ be its dual (of the continuous functionals). If $\cal H$ is a real vector space, I can define: $$\begin{align}\Phi\colon\, &{\cal H} \to ...
1
vote
1answer
41 views

Multiple integration problem with exp [duplicate]

Show that $\int_{-\infty}^{\infty} \exp(-x^2)\, dx = \sqrt{\pi}$. I believe it is necessary to use the Fubini's Theorem and the Change of Variables Theorem. I guess the variable shift function can ...
1
vote
1answer
31 views

How to evaluate Area of $B:= \{(x,y,z) \in A | z \le 1 \}$ with $A:=\{(x,y,z) \in \mathbb{R}^3 | x^2 + y^2 = z\} = \frac{\pi}{6}(5 \sqrt{5}-1) $?

I have following problem: Let $$A:=\{(x,y,z) \in \mathbb{R}^3 | x^2 + y^2 = z\} \\ B:= \{(x,y,z) \in A | z \le 1 \}. $$ Compute the area $\mu_2(B) $. First, I thought $\mu_2(B) $ would just be the ...
0
votes
1answer
21 views

Max/min for functions of two variables and more

For the function $f(x,y)=x^2(e^{-x^2-3x-4y^2})$ find: (1.)The points in $\mathbb{R}^2$,where $f(x,y)$ has local extrema. (2.)The biggest and the smallest values of $f(x,y)$ in the set ...
0
votes
0answers
18 views

Is there a $x(t):[a,b]\to \mathbb{R}$ such that $x(t)D(t,\lambda)\geq 0$?

Let $G\neq \emptyset$ be a set, and $\lambda\in G$. Define $D(t,\lambda):[a,b]\to \mathbb{R}$, continuous. My queston is: is there a function $x:[a,b]\to \mathbb{R}$, such that: $x$ is continuous ...
2
votes
1answer
39 views

Continuous function approximated by a polynomial

I have to prove that: If $f$ is a real valued continuous function on the closed interval $[a,b]$ then given $\varepsilon>0$ there is a polynomial $p(x)$ such that $p(a)=f(a)$, $p'(a)= 0$ ...
1
vote
0answers
24 views

Question on Borel functional calculus

I'm studying right now spectral theory of unbounded self-adjoint operators. A corollary of spectral theorem states the following: let $H$ be a (separable) Hilbert space and $(D_T, T)$ a self-adjoint ...
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0answers
34 views

How to prove that: $\exists C>1$ such that $C||g(y)||\geq ||y||$?

I'm proving the following: Let $E,F$ be two Banach spaces; let $f$ be a function $f:E\to F\ $ linear, and such that: "for every sequence $(x_n)_{n\in\mathbb{N}}\subseteq E$ which converges to ...
3
votes
0answers
50 views

Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$

Assume $\Omega$ is open bounded domain in $\mathbb R^n$ Is $C^\infty_{0}(\bar{\Omega})$ dense in the hilbert space $W^{2,2}(\Omega)\cap W^{1,2}_{0}(\Omega)$ with inner product ...
2
votes
1answer
70 views

Looking for Advice Self Study Analysis [closed]

$\space$ This summer I have some spare time and I was wanting to dedicate some time to self studying some more math. The reasons are many, but mostly because I am wanting to be best prepared for my ...
1
vote
2answers
45 views

Show that $\sup(\frac{1}{A})=\frac{1}{\inf A}$

Given nonempty set $A$ of positive real numbers, and define $$\frac{1}{A}=\left\{z=\frac{1}{x}:x\in A \right\}$$ Show that $$\sup\left(\frac{1}{A}\right)=\frac{1}{\inf A}$$ let ...
1
vote
2answers
34 views

Continuous version of the Cantor-Schroder-Bernstein Theorem [duplicate]

Does the existence of continuous injections $f: A\rightarrow B$ and $g: B\rightarrow A$ imply the existence of a bicontinuous bijection between A and B (ie topological equivalence)? If not, what is a ...
0
votes
4answers
73 views

The sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$

What is the sum of the series $\sum_{n=0}^\infty\left(\frac{4n+3}{5^n}\right)$ ? I got that the series converges and the sum seems to be $5$. When trying to explicitly get the sum, I tried to find the ...
0
votes
1answer
63 views

Riemann Integral

I tried to do the following excercise Let $a,b \in \mathbb{R}, a<b$. We have a bounded function $ f: [a,b] \rightarrow \mathbb{R}$ which has an integral or in other words, there exists a ...
1
vote
1answer
34 views

(Locally) sym., homogenous spaces and space forms

We had some definitions of particular types of Riemannian manifolds in our lecture 1.) Locally symmetric spaces. They were Riemannian manifolds with the property that $\nabla R=0$ everywhere. 2.) ...
0
votes
0answers
18 views

Problem in series expansion

I have some problem in understanding this text from The Princeton Companion to Mathematics: Suppose that one seeks an expansion of $f(x) = \sqrt{x + i}$ in powers of $i$. In general, only integer ...
0
votes
1answer
27 views

show the following including euclidean norm

How do I show that the open set is the same as when we make the definition of an open set in $R^n$ using norm a.$||.||_1$ b.$||.||_{\infty}$ c.$||.||_2$(Euclidean norm) So I know that for $V ...
3
votes
1answer
25 views

Example of non-commutative infinite product of complex numbers.

I have read a proof of the following theorem in Rudin's Real and Complex Analysis: Suppose $\{u_n\}$ is a sequence of bounded complex functions on a set S, such that $\sum |u_n(s)|$ I converges ...
0
votes
1answer
46 views

Proving a differential inequality without performing iteration

I'm seeking a better proof of the following fact: If $g$ is a non-negative bounded function, $g(0)=0$ and $g'(t)\leq \sqrt{g(t)}$ for all $t>0$, then $g(t)\leq t^2/4$. The upper bound $t^2/4$ is ...